What is this arithmetic function?

Question

What is the name of the following "prime partition" function $\nu : \mathbb{N} \rightarrow \mathbb{N}$?

$\nu(n)$ = number of distinct ways of writing $n$ as a sum of prime numbers

For example:

$\nu(1) = 0$

$\nu(2) = 1 = \nu(3) = \nu(4)$

$\nu(5) = 2$

Has it been investigated somewhere - its generating function in particular?

Ditto for the companion function $\nu_1$ which allows 1 to appear as an addend as well. It grows much faster than $\nu$, of course.

Answer 1

See Vaughan: On the number of partitions into primes, Ramanujan J. 15 (2008), 109-121. See also this corrigendum and the related post Wrong asymptotics of OEIS A000607 (number of partitions of an integer in prime parts)?.